Note that \(E_{\lbrace \mu -\rho _n^{(j)}\rbrace }\)  is the PRV component [1]} of the tensor product of \(E_{\mu }\)  with the contragredient \(\mathfrak {k}\) -type of \(E_{\rho _n^{(j)}}\)  (which is just \(E_{\rho _n^{(j)}}\)  in the current setting).
